Problem 14 - Beginning and Intermediate Algebra

Question

The following conversation took place between one of the authors of this text and his sor Jack, when Jack was 4 years old. DADDY: "Jack, what is \(3+0 ? "\) JACK: \(\quad\) "3." DADDY: "Jack, what is \(4+0 ? "\) JACK: \(\quad\) "4. And Daddy, string plus zero equals string!" What property of addition did Jack recognize?

Step-by-Step Solution

Answer

The Identity Property of Addition.

Step 1: Understand the Addition Problem

In the conversation, Jack is asked to add 0 to different numbers. Notice that Jack correctly identifies the sum in both cases: 3 + 0 = 3 and 4 + 0 = 4.

Step 2: Identify the Consistent Pattern

Jack observes that adding 0 to any number does not change the value of that number. He generalized this concept by stating 'string plus zero equals string,' meaning any value plus zero remains that value.

Step 3: Recognize the Property

The property Jack recognized is that adding zero to any number does not change the number. This is known as the Identity Property of Addition.

Key Concepts

addition

Addition is one of the basic operations in arithmetic. It combines two or more numbers to get a total or sum. For example, if we add 2 and 3, we get 5: 2 + 3 = 5.

Addition is widely used in daily life, whether you're adding money, counting objects, or calculating time.

It's important to know that addition is commutative, meaning that the order of the numbers doesn't change the sum. For instance, 3 + 2 = 5 and 2 + 3 = 5.

Another property of addition is associativity. This means that when adding three or more numbers, the grouping doesn't affect the sum: (1 + 2) + 3 = 1 + (2 + 3).

identity property

The identity property of addition states that any number plus zero equals the same number. For example, 7 + 0 = 7.

This property helps maintain the value of the original number when zero is added. It is a fundamental concept in arithmetic and algebra.

Jack recognized this property in the exercise. When asked what is 3 + 0, he correctly said 3. This shows that adding zero doesn't change the value. This is a key observation, especially for young learners.

The identity property also extends to algebraic expressions. For example, if we have a variable and add zero, the value stays the same: x + 0 = x.

elementary algebra

Elementary algebra is the foundation of algebra. It includes understanding and using variables, constants, and basic operations like addition and subtraction.

Variables in algebra represent unknown values and are usually denoted by letters like x or y. Constants have fixed values such as numbers.

When dealing with the identity property of addition, you often use variables. For example, if you have the expression x + 0, the value will still be x.

This is because of the identity property.

Understanding elementary algebra concepts like the identity property helps students build a strong mathematical foundation. These basics are crucial for more advanced math topics later on.